11

Lab FiveLab Five

22

Lessons to be LearnedLessons to be Learned ““Look before you leap”Look before you leap”

Get a feel for the data using graphical techniques, i.e. Get a feel for the data using graphical techniques, i.e. exploratory data analysisexploratory data analysis

In statistics, we do not what the “truth” is, so In statistics, we do not what the “truth” is, so keep an open mindkeep an open mind Try different models, e.g. if linear does not work, try Try different models, e.g. if linear does not work, try

log-loglog-log

Shifting the regression line by shifting the Shifting the regression line by shifting the intercept if the data may fall into different classesintercept if the data may fall into different classes

33

Lab 5 Data : “Fortune 500” Top 50Lab 5 Data : “Fortune 500” Top 50Rank Company Industry Revenue $M

1 General Motors Motor Vehicles and Parts 189058

2 Wal-Mart Stores General Merchandisers 166809

3 Exxon Mobil Petroleum Refining 163881

4 Ford Motor Motor Vehicles and Parts 162558

5 General Electric Diversified Financials 111630

6 Intl. Business Machines Computers, Office Equipment 87548

7 Citigroup Diversified Financials 82005

8 AT&T Telecommunications 62391

9 Philip Morris Tobacco 61751

10 Boeing Aerospace 57993

11 Bank of America Corp. Commercial banks 51392

12 SBC Communications Telecommunications 49489

13 Hewlett-Packard Computers, Office Equipment 48253

14 Kroger Food and Drug Stores 45351.62

15 State Farm Insurance Cos Insurance; P&C(mutual) 44637.25

16 Sears Roebuck General Merchandisers 41071

17 American International Group Insurance; P&C(stock) 40656.08

18 Enron Pipelines 40112

19 TIAA-CREF Insurance: Life, Health(mutual) 39410.2

20 Compaq Computer Computers, Office Equipment 38525

44

Histogram of Revenue, top 50 Fortune 500 1999

05

1015202530354045

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Exploratory Data AnalysisExploratory Data Analysis

55

Exploratory Data AnalysisExploratory Data Analysis

BoxPlot

25986 75986 125986 175986

Smallest = 25986

Q1 = 30133.975

Median = 37044.3

Q3 = 48562

Largest = 189058

IQR = 18428.025

Outliers: 189058, 166809, 163881, 162558, 111630, 87548, 82005,

GMcitigroupAT&T

Aetna

66

Plot of Assets Vs. RevenuePlot of Assets Vs. RevenueAssets Vs. Revenue Millions $, Fortune 500 1999, Top 50

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Revenue, Millions $

Ass

ets,

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Dependent Variable: ASSETSMethod: Least Squares

Sample: 1 50Included observations: 50

Variable Coefficient Std. Error t-Statistic Prob.

REVENUE 1.067927 0.614944 1.736625 0.0889C 83847.72 39395.83 2.128340 0.0385

R-squared 0.059116 Mean dependent var 138113.5Adjusted R-squared 0.039514 S.D. dependent var 173101.5S.E. of regression169647. Akaike info criterion 26.96001Sum squared resid 1.38E+1 Schwarz criterion 27.03649Log likelihood -672.0001 F-statistic 3.015865Durbin-Watson stat 1.952661 Prob(F-statistic) 0.088868

99

Histogram of ln Revenue. Millions $

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Exploratory data Analysis

1010

Fortune 500, 1999: Assets Vs. Revenue, In Logs

General Electric

CitigroupBank of AmericaFannie May

Chase ManhattenMorgan Stanley

Merrill LynchPrudential

Bank One American InternationalTIAA-CREF

State Farm

Allstate

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Log Revenue

Lo

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Transformation: Ln Assets = a+b Ln Revenue + e

1111

Finance Vs. Trade?Finance Vs. Trade?Fortune 500, 1999: Assets Vs. Revenue, In Logs

General Motors

Exxon Mobil

Wal-Mart

Kroger

Ingram Micro

Costco Wholesale

McKesson HBOC

General Electric

CitigroupBank of AmericaFannie May

Chase ManhattenMorgan Stanley

Merrill LynchPrudential

Bank One American InternationalTIAA-CREF

State Farm

Allstate

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1212

1313

Dependent Variable: LNASSETSMethod: Least Squares

Sample: 1 50Included observations: 50

Variable Coefficient Std. Error t-Statistic Prob.

LNSALES 0.919950 0.319084 2.883096 0.0059C 1.294185 3.406777 0.379885 0.7057

R-squared 0.147610 Mean dependent var 11.10485Adjusted R-squared 0.12985 S.D. dependent var 1.243987S.E. of regression1.160413 Akaike info criterion 3.174607Sum squared resid 64.6347 Schwarz criterion 3.251088Log likelihood -77.36517 F-statistic 8.312240Durbin-Watson stat 1.70002 Prob(F-statistic) 0.005877

1414

Table 1: Industry and Number of Firms

Industry# of Firms

Aerospace 1

Chemicals 1

Commercial Banks 3

Computers, Office Equipment 3

Diversified Financials 3

Electronics, Electrical Equipment

1

Entertainment 1

Food and Drug Stores 3

General Merchandisers 5

Health Care 1

Insurance 5

Mail, Package, Freight Delivery 1

Motor Vehicles and Parts 2

Network Communications 1

Petroleum Refining 3

Pharmaceuticals 2

Pipelines 1

Securities 2

Semiconductors 1

Soaps, Cosmetics 1

Specialty Retailers 2

Telecommunications 4

Tobacco 1

Wholesalers 2

Industry # of Firms

Aerospace 1

Chemicals 1

Commercial Banks 3

Computers, Office Equipment 3

Diversified Financials 3

Electronics, Electrical Equipment

1

Entertainment 1

Food and Drug Stores 3

General Merchandisers 5

Health Care 1

Insurance 5

Mail, Package, Freight Delivery 1

Motor Vehicles and Parts 2

Network Communications 1

Petroleum Refining 3

Pharmaceuticals 2

Pipelines 1

Securities 2

Semiconductors 1

Soaps, Cosmetics 1

Specialty Retailers 2

Telecommunications 4

Tobacco 1

Wholesalers 2

Industry # of Firms

Aerospace 1

Chemicals 1

Commercial Banks 3

Computers, Office Equipment 3

Diversified Financials 3

Electronics, Electrical Equipment

1

Entertainment 1

Food and Drug Stores 3

General Merchandisers 5

Health Care 1

Insurance 5

Mail, Package, Freight Delivery 1

Motor Vehicles and Parts 2

Network Communications 1

Petroleum Refining 3

Pharmaceuticals 2

Pipelines 1

Securities 2

Semiconductors 1

Soaps, Cosmetics 1

Specialty Retailers 2

Telecommunications 4

Tobacco 1

Wholesalers 2

1515

Ln-ln Regression with Industry Ln-ln Regression with Industry DummiesDummies

1616

Log likelihood

-0.798333

F-statistic

25.09152Durbin-Watson stat

2.427065

Prob(F-statistic)

0.000000

1717

1818

1919

Wald Test: Null HypothesisWald Test: Null Hypothesis

2020

Wald Test Results: reject NullWald Test Results: reject Null

Wald Test:Equation: UntitledNull Hypothesis: C(3)=C(18)

C(6)=C(18)C(12)=C(18)

F-statistic 3.488804 Probability 0.030516Chi-square 10.46641 Probability 0.014990

2121

Wald Test: drop insurance from GroupWald Test: drop insurance from Group

2222

Wald Test results: Accept NullWald Test results: Accept Null

Wald Test:Equation: UntitledNull Hypothesis: C(3)=C(18)

C(6)=C(18)F-statistic 0.300921 Probability 0.742780

Chi-square 0.601842 Probability 0.740136

2323

Wald Test: Equivalent to a Likelihood Wald Test: Equivalent to a Likelihood Ratio testRatio test

Equation with all the dummiesEquation with all the dummies RR2 2 = 0.9601= 0.9601 SSR = 3.022SSR = 3.022 Ln Likelihood = - 0.798333Ln Likelihood = - 0.798333 Estimate 25 regression parametersEstimate 25 regression parameters

Equation with Finance Group Dummy replacing Equation with Finance Group Dummy replacing banks, divfinanc, and securitiesbanks, divfinanc, and securities RR2 2 = 0.9592= 0.9592 SSR = 3.095SSR = 3.095 Ln Likelihood = - 1.393045Ln Likelihood = - 1.393045 Estimate 23 regression parametersEstimate 23 regression parameters

2424

Likelihood Ratio TestLikelihood Ratio Test

Likelihood ratio = Likelihood ratio = λλ = Likelihood = Likelihood (constrained)/Likelihood(unconstrained)(constrained)/Likelihood(unconstrained)

Where -2 lnWhere -2 lnλλ is distributed as Chi square is distributed as Chi square -2 ln -2 ln λλ = -2 [ln Lik(const) – ln Lik(unconst) = -2 [ln Lik(const) – ln Lik(unconst)=2[ln Lik(unconst) –ln Lik(const)]=2[ln Lik(unconst) –ln Lik(const)]=2[-0.798333 – (-1.393045)] =2(0.594712)=2[-0.798333 – (-1.393045)] =2(0.594712) -2ln -2ln λλ = 1.189424 = 1.189424

2525

2626

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Chi Square Test 2 Degrees of FreedomChi Square Test 2 Degrees of Freedom

5.99

5 %

1.19 Our Chi Square statistic

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F-test statisticF-test statisticExplained sum of squares from the banks, Explained sum of squares from the banks,

divfinance and securities dummies equals SSR divfinance and securities dummies equals SSR (regression #2) – SSR(regression #1) = 3.095 – (regression #2) – SSR(regression #1) = 3.095 – 3.022 = 0.0733.022 = 0.073

Degrees of freedom 2Degrees of freedom 2Explained mean square = 0.073/2 = 0.0365Explained mean square = 0.073/2 = 0.0365Unexplained mean square from regression #1 = Unexplained mean square from regression #1 =

0.0322/(n-k) = 0.0322/250.0322/(n-k) = 0.0322/25FF2, 25 2, 25 = 0.0365/0.12088 =0.302= 0.0365/0.12088 =0.302

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FF2, 232, 23 Test; accept null Test; accept null

c(3)=c(6)=c(18)c(3)=c(6)=c(18)

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2929

Wald Test results: Accept NullWald Test results: Accept Null

Wald Test:Equation: UntitledNull Hypothesis: C(3)=C(18)

C(6)=C(18)F-statistic 0.300921 Probability 0.742780

Chi-square 0.601842 Probability 0.740136

3030

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